Competition Vehicle Based Intake Manifold Design
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The Grassmann Manifold
The Grassmann Manifold 1. For vector spaces V and W denote by L(V; W ) the vector space of linear maps from V to W . Thus L(Rk; Rn) may be identified with the space Rk£n of k £ n matrices. An injective linear map u : Rk ! V is called a k-frame in V . The set k n GFk;n = fu 2 L(R ; R ): rank(u) = kg of k-frames in Rn is called the Stiefel manifold. Note that the special case k = n is the general linear group: k k GLk = fa 2 L(R ; R ) : det(a) 6= 0g: The set of all k-dimensional (vector) subspaces ¸ ½ Rn is called the Grassmann n manifold of k-planes in R and denoted by GRk;n or sometimes GRk;n(R) or n GRk(R ). Let k ¼ : GFk;n ! GRk;n; ¼(u) = u(R ) denote the map which assigns to each k-frame u the subspace u(Rk) it spans. ¡1 For ¸ 2 GRk;n the fiber (preimage) ¼ (¸) consists of those k-frames which form a basis for the subspace ¸, i.e. for any u 2 ¼¡1(¸) we have ¡1 ¼ (¸) = fu ± a : a 2 GLkg: Hence we can (and will) view GRk;n as the orbit space of the group action GFk;n £ GLk ! GFk;n :(u; a) 7! u ± a: The exercises below will prove the following n£k Theorem 2. The Stiefel manifold GFk;n is an open subset of the set R of all n £ k matrices. There is a unique differentiable structure on the Grassmann manifold GRk;n such that the map ¼ is a submersion. -
Mean Value Modelling of a Poppet Valve EGR-System
Mean value modelling of a poppet valve EGR-system Master’s thesis performed in Vehicular Systems by Claes Ericson Reg nr: LiTH-ISY-EX-3543-2004 14th June 2004 Mean value modelling of a poppet valve EGR-system Master’s thesis performed in Vehicular Systems, Dept. of Electrical Engineering at Linkopings¨ universitet by Claes Ericson Reg nr: LiTH-ISY-EX-3543-2004 Supervisor: Jesper Ritzen,´ M.Sc. Scania CV AB Mattias Nyberg, Ph.D. Scania CV AB Johan Wahlstrom,¨ M.Sc. Linkopings¨ universitet Examiner: Associate Professor Lars Eriksson Linkopings¨ universitet Linkoping,¨ 14th June 2004 Avdelning, Institution Datum Division, Department Date Vehicular Systems, Dept. of Electrical Engineering 14th June 2004 581 83 Linkoping¨ Sprak˚ Rapporttyp ISBN Language Report category — ¤ Svenska/Swedish ¤ Licentiatavhandling ISRN ¤ Engelska/English ££ ¤ Examensarbete LITH-ISY-EX-3543-2004 ¤ C-uppsats Serietitel och serienummer ISSN ¤ D-uppsats Title of series, numbering — ¤ ¤ Ovrig¨ rapport ¤ URL for¨ elektronisk version http://www.vehicular.isy.liu.se http://www.ep.liu.se/exjobb/isy/2004/3543/ Titel Medelvardesmodellering¨ av EGR-system med tallriksventil Title Mean value modelling of a poppet valve EGR-system Forfattare¨ Claes Ericson Author Sammanfattning Abstract Because of new emission and on board diagnostics legislations, heavy truck manufacturers are facing new challenges when it comes to improving the en- gines and the control software. Accurate and real time executable engine models are essential in this work. One successful way of lowering the NOx emissions is to use Exhaust Gas Recirculation (EGR). The objective of this thesis is to create a mean value model for Scania’s next generation EGR system consisting of a poppet valve and a two stage cooler. -
On Manifolds of Tensors of Fixed Tt-Rank
ON MANIFOLDS OF TENSORS OF FIXED TT-RANK SEBASTIAN HOLTZ, THORSTEN ROHWEDDER, AND REINHOLD SCHNEIDER Abstract. Recently, the format of TT tensors [19, 38, 34, 39] has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor U ∈ Rn1×...×nd and for the manifold of tensors of TT-rank r. As a first result, we prove that the TT (or compression) ranks ri of a tensor U are unique and equal to the respective seperation ranks of U if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that d the set T of TT tensors of fixed rank r forms an embedded manifold in Rn , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices [7], we introduce certain gauge conditions to obtain a unique representation of the tangent space TU T of T and deduce a local parametrization of the TT manifold. The parametrisation of TU T is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems [33]. We conclude with remarks on those applications and present some numerical examples. 1. Introduction The treatment of high-dimensional problems, typically of problems involving quantities from Rd for larger dimensions d, is still a challenging task for numerical approxima- tion. This is owed to the principal problem that classical approaches for their treatment normally scale exponentially in the dimension d in both needed storage and computa- tional time and thus quickly become computationally infeasable for sensible discretiza- tions of problems of interest. -
Engine Block Materials and Its Production Processes
ENGINE BLOCK MATERIALS AND ITS PRODUCTION PROCESSES 2.2 THE CAST IRON MONOLITHIC BLOCK The widespread use of cast iron monolithic block is as a result of its low cost and its formidability. This type of block normally comes as the integral type where the engine cylinder and the upper crankcase are joined together as one. The iron used for this block is the gray cast iron having a pearlite-microstructure. The iron is called gray cast iron because its fracture has a gray appearance. Ferrite in the microstructure of the bore wall should be avoided because too much soft ferrite tends to cause scratching, thus increasing blow-by. The production of cast iron blocks using a steel die is rear because its lifecycle is shortened as a result of the repeated heat cycles caused by the molten iron. Sand casting is the method widely used in the production of cast iron blocks. This involves making the mould for the cast iron block with sand. The preparation of sand and the bonding are a critical and very often rate-controlling step. Permanent patterns are used to make sand molds. Usually, an automated molding machine installs the patterns and prepares many molds in the same shape. Molten metal is poured immediately into the mold, giving this process very high productivity. After solidification, the mold is destroyed and the inner sand is shaken out of the block. The sand is then reusable. The bonding of sand is done using two main methods: (i) the green sand mold and (ii) the dry sand mold. -
INTRODUCTION to ALGEBRAIC GEOMETRY 1. Preliminary Of
INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1. Preliminary of Calculus on Manifolds 1.1. Tangent Vectors. What are tangent vectors we encounter in Calculus? 2 0 (1) Given a parametrised curve α(t) = x(t); y(t) in R , α (t) = x0(t); y0(t) is a tangent vector of the curve. (2) Given a surface given by a parameterisation x(u; v) = x(u; v); y(u; v); z(u; v); @x @x n = × is a normal vector of the surface. Any vector @u @v perpendicular to n is a tangent vector of the surface at the corresponding point. (3) Let v = (a; b; c) be a unit tangent vector of R3 at a point p 2 R3, f(x; y; z) be a differentiable function in an open neighbourhood of p, we can have the directional derivative of f in the direction v: @f @f @f D f = a (p) + b (p) + c (p): (1.1) v @x @y @z In fact, given any tangent vector v = (a; b; c), not necessarily a unit vector, we still can define an operator on the set of functions which are differentiable in open neighbourhood of p as in (1.1) Thus we can take the viewpoint that each tangent vector of R3 at p is an operator on the set of differential functions at p, i.e. @ @ @ v = (a; b; v) ! a + b + c j ; @x @y @z p or simply @ @ @ v = (a; b; c) ! a + b + c (1.2) @x @y @z 3 with the evaluation at p understood. -
Section 02 - Block Basics
Block Basics – Section 2 Section 02 - Block Basics 2.0 Small Block 330 & 350 Block Key Differences. The key differences between the 330 and 350 are the 350’s larger bore and the Generation 1 Cast Iron Small Block V-8 Facts 330’s forged crank. General. In 1964 Olds replaced their small block 215 V8 with 1964 – 1966 Valve Lifter Angle. All 1964–1966 blocks used a a cast iron block of completely new design. The 330 V-8 different valve lifter angle of attack on the cam (45). Thus shared none of its engine block architecture with that of the 1964–1966 330 blocks CANNOT USE 1967 AND LATER 215 V-8 and the 225 V-6 sourced from Buick. The engine CAMS. All 1964–1966 cams WILL NOT WORK in 1967 and was no longer aluminum, but cast iron, as weight became later blocks. Later blocks used a 39 lifter angle. Blocks with less of a factor with the engine going into both the larger a “1” or “1A” cast up near the oil filler tube used the 45 lifter mid-sized F-85s, Cutlasses and the full-size Jetstars angle and should be avoided, if possible. introduced in that year. The engine was designed as a replacement for the 215, but was cast iron and enlarged in Early 330 Rocker Arms. The first run of 330 blocks was anticipation of the growth in size of the mid-size cars, where equipped with rocker arms similar to the previous 394 block it was to be primarily used and as the workhorse for the that traces its heritage back to 1949. -
DIFFERENTIAL GEOMETRY COURSE NOTES 1.1. Review of Topology. Definition 1.1. a Topological Space Is a Pair (X,T ) Consisting of A
DIFFERENTIAL GEOMETRY COURSE NOTES KO HONDA 1. REVIEW OF TOPOLOGY AND LINEAR ALGEBRA 1.1. Review of topology. Definition 1.1. A topological space is a pair (X; T ) consisting of a set X and a collection T = fUαg of subsets of X, satisfying the following: (1) ;;X 2 T , (2) if Uα;Uβ 2 T , then Uα \ Uβ 2 T , (3) if Uα 2 T for all α 2 I, then [α2I Uα 2 T . (Here I is an indexing set, and is not necessarily finite.) T is called a topology for X and Uα 2 T is called an open set of X. n Example 1: R = R × R × · · · × R (n times) = f(x1; : : : ; xn) j xi 2 R; i = 1; : : : ; ng, called real n-dimensional space. How to define a topology T on Rn? We would at least like to include open balls of radius r about y 2 Rn: n Br(y) = fx 2 R j jx − yj < rg; where p 2 2 jx − yj = (x1 − y1) + ··· + (xn − yn) : n n Question: Is T0 = fBr(y) j y 2 R ; r 2 (0; 1)g a valid topology for R ? n No, so you must add more open sets to T0 to get a valid topology for R . T = fU j 8y 2 U; 9Br(y) ⊂ Ug: Example 2A: S1 = f(x; y) 2 R2 j x2 + y2 = 1g. A reasonable topology on S1 is the topology induced by the inclusion S1 ⊂ R2. Definition 1.2. Let (X; T ) be a topological space and let f : Y ! X. -
Manifold Reconstruction in Arbitrary Dimensions Using Witness Complexes Jean-Daniel Boissonnat, Leonidas J
Manifold Reconstruction in Arbitrary Dimensions using Witness Complexes Jean-Daniel Boissonnat, Leonidas J. Guibas, Steve Oudot To cite this version: Jean-Daniel Boissonnat, Leonidas J. Guibas, Steve Oudot. Manifold Reconstruction in Arbitrary Dimensions using Witness Complexes. Discrete and Computational Geometry, Springer Verlag, 2009, pp.37. hal-00488434 HAL Id: hal-00488434 https://hal.archives-ouvertes.fr/hal-00488434 Submitted on 2 Jun 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Manifold Reconstruction in Arbitrary Dimensions using Witness Complexes Jean-Daniel Boissonnat Leonidas J. Guibas Steve Y. Oudot INRIA, G´eom´etrica Team Dept. Computer Science Dept. Computer Science 2004 route des lucioles Stanford University Stanford University 06902 Sophia-Antipolis, France Stanford, CA 94305 Stanford, CA 94305 [email protected] [email protected] [email protected]∗ Abstract It is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling conditions. In this paper, we prove that these results do not extend to higher-dimensional manifolds, even under strong sampling conditions such as uniform point density. -
Hodge Theory
HODGE THEORY PETER S. PARK Abstract. This exposition of Hodge theory is a slightly retooled version of the author's Harvard minor thesis, advised by Professor Joe Harris. Contents 1. Introduction 1 2. Hodge Theory of Compact Oriented Riemannian Manifolds 2 2.1. Hodge star operator 2 2.2. The main theorem 3 2.3. Sobolev spaces 5 2.4. Elliptic theory 11 2.5. Proof of the main theorem 14 3. Hodge Theory of Compact K¨ahlerManifolds 17 3.1. Differential operators on complex manifolds 17 3.2. Differential operators on K¨ahlermanifolds 20 3.3. Bott{Chern cohomology and the @@-Lemma 25 3.4. Lefschetz decomposition and the Hodge index theorem 26 Acknowledgments 30 References 30 1. Introduction Our objective in this exposition is to state and prove the main theorems of Hodge theory. In Section 2, we first describe a key motivation behind the Hodge theory for compact, closed, oriented Riemannian manifolds: the observation that the differential forms that satisfy certain par- tial differential equations depending on the choice of Riemannian metric (forms in the kernel of the associated Laplacian operator, or harmonic forms) turn out to be precisely the norm-minimizing representatives of the de Rham cohomology classes. This naturally leads to the statement our first main theorem, the Hodge decomposition|for a given compact, closed, oriented Riemannian manifold|of the space of smooth k-forms into the image of the Laplacian and its kernel, the sub- space of harmonic forms. We then develop the analytic machinery|specifically, Sobolev spaces and the theory of elliptic differential operators|that we use to prove the aforementioned decom- position, which immediately yields as a corollary the phenomenon of Poincar´eduality. -
Appendix D: Manifold Maps for SO(N) and SE(N)
424 Appendix D: Manifold Maps for SO(n) and SE(n) As we saw in chapter 10, recent SLAM implementations that operate with three-dimensional poses often make use of on-manifold linearization of pose increments to avoid the shortcomings of directly optimizing in pose parameterization spaces. This appendix is devoted to providing the reader a detailed account of the mathematical tools required to understand all the expressions involved in on-manifold optimization problems. The presented contents will, hopefully, also serve as a solid base for bootstrap- ping the reader’s own solutions. 1. OPERATOR DEFINITIONS In the following, we will make use of some vector and matrix operators, which are rather uncommon in mobile robotics literature. Since they have not been employed throughout this book until this point, it is in order to define them here. The “vector to skew-symmetric matrix” operator: A skew-symmetric matrix is any square matrix A such that AA= − T . This implies that diagonal elements must be all zeros and off-diagonal entries the negative of their symmetric counterparts. It can be easily seen that any 2× 2 or 3× 3 skew-symmetric matrix only has 1 or 3 degrees of freedom (i.e. only 1 or 3 independent numbers appear in such matrices), respectively, thus it makes sense to parameterize them as a vector. Generating skew-symmetric matrices ⋅ from such vectors is performed by means of the []× operator, defined as: 0 −x 2× 2: = x ≡ []v × () × x 0 x 0 z y (1) − 3× 3: []v = y ≡ z0 − x × z −y x 0 × The origin of the symbol × in this operator follows from its application to converting a cross prod- × uct of two 3D vectors ( x y ) into a matrix-vector multiplication ([]x× y ). -
United States Patent (19) 11 Patent Number: 5,950,587 Sattler Et Al
USOO5950587A United States Patent (19) 11 Patent Number: 5,950,587 Sattler et al. (45) Date of Patent: Sep. 14, 1999 54 CONTINUOUSLY WARIABLE RUNNER 2682431-A1 4/1993 France. LENGTH MANIFOLD 3825000 A1 2/1989 Germany .......................... 123/184.55 60-224923 11/1985 Japan .................... ... 123/184.55 75 Inventors: Eric R. Sattler, Trenton; Joel S. 2239899 7/1991 United Kingdom ...... ... 123/184.55 Myers, Southgate; Michael J. Haspel, Westland, all of Mich. Primary Examiner Erick R. Solis 73 Assignee: BASF Corporation, Mount Olive, N.J. ASSistant Examiner Brian Hairston Attorney, Agent, or Firm-Ryan W. Massey; James J. Drake 22 Filed: Jul. 22, 1998 A continuously variable runner length manifold is provided (51) Int. Cl. ................................................ FO2B 27/06 for an internal combustion engine. The continuously vari 52 U.S. Cl. .................. 123/184.55; 123/18453 able runner length manifold includes a housing having an 58 Field of Search ........................... 123/184.55, 18453 inlet port and a plurality of outlet ports defined by a plurality of Stacked manifold Sections. A plurality of runner members 56) References Cited are rotatably mounted within the housing. The runner mem bers include wall portions which combine with the housing U.S. PATENT DOCUMENTS to define a plurality of runners in communication with a 4,619,226 10/1986 Ueda et al. ........................ 123s2 MB plenum and a respective one of the outlet ports. The runners 4,699,630 10/1987 Lee et al. ............................... 48/1801 have a length which is continuously variable upon rotation of the runner members relative to the housing. FOREIGN PATENT DOCUMENTS 237755 A2 9/1987 European Pat. -
Compacted Graphite Iron for Diesel Engine Cylinder Blocks
Compacted Graphite Iron for Diesel Engine Cylinder Blocks Wilson Luiz Guesser, Dr Eng - Tupy Fundições and UDESC - Joinville, SC – Brazil – [email protected] Pedro Ventrela Duran, M Sc - Tupy Fundições - Joinville, SC – Brazil – [email protected] Walmor Krause, M Sc - Tupy Fundições - Joinville, SC – Brazil – [email protected] Abstract The recent trends in diesel engines are discussed, and also the consequences in materials selection for engine cylinder blocks. The increasing demand for higher specific power and the need for weight reduction and decrease of emissions require the use of stronger materials for cylinder blocks, opening a promising space for compacted graphite iron (CGI). The properties of CGI are described, in particular those required for engine cylinder blocks (fatigue strength and elastic modulus). Results of mechanical properties from actual castings are presented. As a result of the mechanical properties improvement, some design opportunities are suggested, in order to decrease the weight of the cylinder block. Keywords diesel engines, compacted graphite iron, cylinder blocks 1. Introduction: In a recent paper discussing CGI uses for engine cylinder blocks and heads, S. Dawson (2001) stated that “the Iron Age is just beginning” and we have taken this statement for the present paper. His point of view, added to those from Rizzo (2001), discloses the enormous CGI potential for manufacturing engine components, especially for diesel engines. The growth of diesel engines usage for passenger cars has been remarkable in Europe (see Fig. 1), especially after the introduction of the high-pressure common-rail technology (Buchholz, 2003). This growth is surrounded by requirements as: less fuel consumption, emissions reduction , larger power output and torque, and for passenger cars more compact engines are being required due to space limitations.